Date: Nov 17, 2013 7:22 AM
Author: GS Chandy
Subject: Re: Simplifying Algebraic Expressions with Subtracted Expressions
Pam posted Nov 17, 2013 10:44 AM (http://mathforum.org/kb/thread.jspa?threadID=2606342) - GSC's remarks interspersed:> > > > On Nov 16, 2013, at 4:03 PM, Pam> <Pamkgm@hotmail.com>> > wrote:> > > > > The biggest difference I see is that the algebra> > student may be able to go through the motions> > mechanically, using memorized procedures> > > > Is it possible Pam that you don't know what they> are> > doing in their head, and that it's anything but> > mechanical?> > > > > Bob Hansen> > > > > > You misunderstood. For those of us who did well in> algebra, I would say that most of us had a lot going> on in our heads that was anything but mechanical. >I'd go further: I suggest that ANYONE who has EVER learned ANYTHING has ALWAYS had a lot going on in his/ her/ their head(s) that was anything but mechanical. It is, in my opinion, not possible actually to learn anything '(purely) mechanically'. This is so very obvious to me that I find it astonishing that some people still do insist on 'by-rote learning' by students. The 'by-rote stuff' may well contribute to recall of specific facts, figures and stuff (which may contribute to making that available for application) - but not to real 'learning'.>> In my statement above, I said "may". Unfortunately,> there are students who approach algebra mechanically,> through memorized procedures, because they lack> fundamental understanding. >This is (IMHO) the direct result of poor teaching.>> Also unfortunately, they> think the memorized procedures are the only way to> approach problems, that procedures *are* the algebra,> so to speak. And, no, although they may be able to> solve simple problems that follow exactly what they> understand of the procedure, they are not generally> successful. Prior to remediation, that is.> As I'm not a US-based teacher I don't know how good or effective 'remediation' is in the USA. In fact, I'm not a teacher at all - I'm just a person most keenly interested in the 'learning process' - which I claim is the only hope for humanity as a whole. Most of our planet's issues and problems derive from the fact that we (humanity) have not 'learned effectively' (though we've always had the opportunity to learn).>> As Wayne says, this won't happen with a solid> grounding, but I think the relatively small> percentage who get that solid grounding in the US had> to do so mostly on their own. For the reasons that> Liping Ma has found in her analyses of US math> education, among others.> > Pam>I don't know about the 'solid grounding' provided in various disciplines/subject areas in various countries. I believe the solid grounding is lacking right through our educational system, in practically all fields, in practically all disciplines, in practically all countries - there are, of course, exceptional teachers everywhere who do manage to apply, effectively, the 'learning+teaching dyad' to their professional work as teachers. It is my claim that we'd have a great many more effective teachers if 'teaching' were, in practice, considered a part of the 'learning+teaching dyad' (and not a 'thing-in-itself' as it is too often). A little practical understanding of 'systems' by teachers would help significantly.